Subshifts from sofic shifts and Dyck shifts, zeta functions and topological entropy

0.19 MB·English

by  Kokoro Inoue

SUBSHIFTS FROM SOFIC SHIFTS AND DYCK SHIFTS,
ZETA FUNCTIONS AND TOPOLOGICAL ENTROPY
0
1 KOKOROINOUEANDWOLFGANGKRIEGER
0
2
Abstract. Weintroduceaclassofcodedsystemsthatweconstructfromsofic

n systems and Dyck shifts and we study a class of subshifts that we obtain by
a excluding words of length two from Dyck shifts. We derive expressions for
J zeta functions and topological entropy. We derive an expression for the zeta
2 function of certain subshifts that we obtain by excluding words from Dyck
1 shifts and of certain subshifts that we obtain by excluding words from the
subshiftsthatareconstructed fromfullshiftsandDyckshifts.

]
S
D
Keywords: subshift, zeta function, topological entropy, sofic shift, Dyck shift,
.
h Motzkin shift, Schr¨oder shift, circular code
t AMS Subject Classification: primary 37B10, secondary 05A15
a
m
1. Introduction
[
Let Σ be a finite alphabet. On ΣZ there acts the shift that sends the point
1
v (xi)i∈Z ∈ ΣZ into the point (xi+1)i∈Z ∈ ΣZ. For a shift invariant set Y ⊂ ΣZ we
9 denote the set of periodic points in Y by P(Y) and we denote by Pn(Y) the set of
3 p∈P(Y)thathaveperiodn∈N. ThezetafunctionofashiftinvariantsetY ⊂ΣZ
8
is defined by
1
. ζY(z)=eΣn∈Nn1card(Pn(Y))zn.
1
0 The dynamical systems that are given by the closed shift invariant subsets of ΣZ,
0 with the restriction of the shift acting on them, are called subshifts. These are
1 studied in symbolic dynamics. For an introduction to symbolic dynamics see [Ki]
: Z
v or[LM]. AwordiscalledadmissibleforasubshiftX ⊂Σ ifitappearssomewhere
i in a point of X. We denote the languageof admissible wordsof a subshift X ⊂ΣZ
X
by L(X), and we denote the set of words in L(X) of length n by L (X),n ∈ N.
n
ar The n-block system of the subshift X ⊂ ΣZ is a subshift with alphabet Ln(X), a
topological conjugacy of the subshift onto its n-block system being given by the
map
x→((xj)i<j≤i+n)i∈Z (x∈X).
The topological entropy of a subshift X ⊂ΣZ is given by
1
h(X)= lim logcard(L (X)).
n
n→∞n
The edge shift of finite directed graph has as its language of admissible words
the set of finite paths in the graph. With the adjacency matrix A of the directed
graphthe zetafunctionofthe edgeshiftofthegraphisgivenby 1 (see e.g.
det(1−Az)
[LM, Theorem 6.4.6]). A subshift X ⊂ ΣZ is said to be of finite type, if there is a
finite set F of words in the symbols of the alphabet Σ such that X is equal to the
Z
setofpoints inΣ inwhichno wordinF appears. Sofic systems[W; LM,Chapter
1
2 KOKOROINOUEANDWOLFGANGKRIEGER
3]aretheimagesofsubshiftsoffinitetypeundercontinuousshiftcommutingmaps.
The zeta function of sofic shifts is rational [LM, Theorem 6.4.8; BR, Section 3]. A
finite directed labeled graph, in which every vertex has at least one incoming edge
andat least one outgoing edge, presents a sofic system whose admissible wordsare
the label sequences of finite paths in the graph. The labeling of a directed graph
with label alphabet Σ is called1-rightresolvingif for every σ ∈Σ every vertex has
at most one outgoing edge that carries the label σ. Every topologically transitive
soficsystemiscanonicallypresentedbya1-rightresolvingirreduciblefinitedirected
graph that is known as its right Fischer automaton [F].
For a fomal language L of words in the symbols of the alphabet Σ we denote its
generating function by g(L), and we denote by Y(L) the set of x ∈ ΣZ such that
there are indices i ,k∈Z, such that i <i ,k ∈Z, and
k k k+1
(x ) ∈L, k ∈Z. (1.1)
j ik−1≤j<ik
TheclosureofY(L)is asubshiftthatiscalledthe codedsystemofL[BH]. Acode
in the symbols of the alphabet Σ is said to be circular if for x ∈ P(Y(C)) the set
{i ,k ∈Z} of indices, such that (1.1) holds, is unique ([BP], section VII.1). For a
k
circular code C
1
ζ = ,
Y(C) 1−g(C,z)
(see e.g. [St, Proposition 4.7.11]).
In this paper we construct coded systems that are close to previously described
familiesofcodedsystems[Kr,I,M]. Werecalltherelevantdefinitionsandintroduce
notation. Let Γ be a finite set with more than one element. The set Γ will remain
fixed throughout the paper. We set
N =card(Γ).
We denote the generators of the Dyck inverse monoid D (the polycyclic inverse
N
monoid of [NP]) by γ(−),γ(+),γ ∈Γ. These satisfy the relations
1, if γ =γ′,
γ(−)γ′(+)=
(0, if γ 6=γ′, γ,γ′ ∈Γ.
We recall the construction of the Dyck shifts [Kr] and of the Motzkin shifts [M,
I]. The Dyck shift D is the subshift with alphabet {γ(−),γ(+) : γ ∈ Γ} and
N
admissible words (γ ) ,I ∈N, given by the condition
i 1≤i≤I
γ 6=0. (1.2)
i
1≤i≤I
Y
A word (γ ) ∈L(D ),I ∈N, is called a Dyck word if
i 1≤i≤I N
γ =1. (1.3)
i
1≤i≤I
Y
TheMotzkinshiftM isthesubshiftwithalphabet{γ(−),γ(+):γ ∈Γ}∪{1}and
N
admissible words (γ ) ,I ∈ N, also given by the condition (1.2), the Motzkin
i 1≤i≤I
wordsbeing definedagainby(1.3). TheDyckshiftD canbe viewedasthe coded
N
systemoftheDyckcodewhichisthesetofDyckwordsthatcannotbewrittenasa
non-trivial concatenation of Dyck words. The Motzkin shift M can be viewed as
N
the codedsystemofthe Motzkincode whichis defined asthe setofMotzkinwords
SUBSHIFTS FROM SOFIC SHIFTS AND DYCK SHIFTS 3
that cannot be written as a non-trivial concatenation of Motzkin words minus the
set {1}.
In[HI]anecessaryandsufficientconditionwasgivenforanirreduciblesubshiftof
finitetypetoembedintoaDyckshift,andin[HIK]thiscriterionwasextendedtoa
moregeneralclassof targetshifts that wereconstructedby means ofgraphinverse
semigroups. For the case that the graph inverse semigroup is the Dyck inverse
monoid D ,N > 1, we recall the construction of these target shifts. Denote by
N
D−(D+) the free semigroupwith generatorsα (n) (α (n)), 1≤n≤N. Let there
N N − +
be given a finite irreducible graph with vertex set V and edge set E, together with
a labeling map
λ:E →D− ∪{1}∪D+.
N N
The labeling map extends to paths (e ) in the graph by
i 1≤i≤I
λ((e ) ))= λ(e ).
i 1≤i≤I i
1≤i≤I
Y
Assume that for all V,V′ ∈ V there exists a path b in the graph (V,E) that starts
atV andends atV′ suchthatλ(b)=1. Also assumethat for all V ∈V andfor all
γ ∈ Γ there exist cycles b(−) and b(+) from V to V such that λ(b(−)) = γ(−) and
λ(b(+))=γ(+). The labeleddirectedgraph(V,E,λ) presents a subshiftX(V,E,λ)
that is a subsystem of the edge shift of the graph (V,E) with L(X(V,E,λ)) equal
to the set of paths b in the graph (V,E) such that λ(b)6=0. As in in [HIK] we call
X(V,E,λ) a D -presentation. In this way the Dyck shift D ,N >1, is presented
N N
byagraphwithasinglevertexand2N loopsthatcarrythelabelsγ(−),γ(+),γ ∈Γ.
The graph that presents the Motzkin shift has an additional loop that carries the
label 1. As in [HI] we say that a periodic point p of X(V,E,λ) has a negative
(positive) multiplier if there exists an i ∈ Z such that, with a period π of p, one
has that λ((x ) ) is in D−(D+), and we say that p is neutral if there exists
j i≤j<i+π N N
an i ∈ Z such that λ((x ) ) = 1. One can attempt to construct a D -
j i≤j<i+π N
presentationfromagivenD -presentationX(V,E,λ)byexcludingforsomeK >1
N
suitably chosen words of length K from X(V,E,λ). This amounts to excluding
symbolsfromtheK-blocksystemofX(V,E,λ),which,giventhecorrectlabeling,is
itselfaD -presentation. Providedonehasnotexcludedtoomanywords,oneisleft
N
with a subshift that is still a D -presentation. The examples of D -presentations
N N
that were givenin [HIK] were obtained in this way,and the D -presentationsthat
N
we will encounter in this paper arise in the same way.
Insection2wedescribeaconstructionofsubshiftsthatgeneralizestheconstruc-
tions of the Dyck and Motzkin shifts. Also the Schr¨oder shifts appear here. The
idea of the construction is to attach loops to a vertex of a finite irreducible 1-right
resolving labeled graph, and to have these loops imitate the behavior of the gener-
ators of the Dyck inverse monoid. For instance, the labeled directed graph can be
the Fischer automaton of an irreducible sofic system. In the case that the labeling
of the directed graphis bijective this constructionyields also a D -presentationof
N
the constructed subshift.
After some preparations in section 3, where we consider mappings that assign
to the elements of a finite set non-empty subsets of the set, we study in section
4 the subshifts that are obtained by removing from the Dyck shifts D words
N
of the form γ(+)γ′(+), or, which is equivalent by symmetry, words of the form
γ(−)γ′(−),γ ∈Γ.
4 KOKOROINOUEANDWOLFGANGKRIEGER
In sections 5 we consider two examples where we remove words of length three
fromDyckshiftsandMotzkinshifts. Insection6weconsideranexamplewherewe
removewordsoflengththreefromtheshiftsthatweconstructinsection2fromthe
full shifts and the Dyck shifts. One checks that the 3-block systemof the subshifts
that are constructed in sections 5 and 6 are D -presentations.
N
The aim is in all cases to obtain an expression for the zeta function of the
subshiftsandtodeterminetheirtopologicalentropy. Thezetafuctionweobtainby
the same method as was usedin [Ke] to obtainthe zeta function ofthe Dyck shifts
(see also [I, KM]). In the subshift X ⊂ΣZ one identifies subshifts of finite type or
sofic shifts Y−,Y+ ⊂X, and one identifies circular codes C−,C0,C+ ⊂L(X) such
that
P(X)=P(Y−∪Y+)∪P(Y(C−)∪Y(C+)), (1.4)
P(Y−∪Y+)∩P(Y(C−)∪Y(C+))=∅, (1.5)
Y(C−)∩Y(C+)=Y(C0). (1.6)
Then
ζ ζ ζ ζ
Y− Y(C−) Y(C+) Y+
ζ = . (1.7)
X
ζ ζ
Y−∩Y+ Y(C0)
In case the subshift under consideration is a D -presentation, P(Y(C−))∪P(Y−)
N
coincides with the set of periodic poins of X that are neutral or have a negative
multiplier, and P(Y+)∪P(Y(C+)) coincides with the set of periodic poins of X
that are neutral or have a positive multiplier.
The Schu¨tzenberger method [Sc] (see also [D]) has been for many years a stan-
dard method that has been routinely applied to solve enumeration problems as
theyariseinthe computationofthegeneratingfunctionsofthecircularcodesfrom
which we obtain zeta function and topological entropy of the coded systems that
we consider here. In [KM] the Schu¨tzenberger method was applied in the compu-
tation of zeta functions of a structurally significant class of coded systems, that
includes the Dyck shifts. (Previously, in [Kr] the topological entropy of the Dyck
shiftD ,N >1,hadbeenshowntobelog(N+1)bymethodsfromergodictheory,
N
and in [Ke] the computation of the zeta function of the Dyck shifts was based on
a probabilistic argument.) Zeta function and topological entropy of the Motzkin
shiftswereobtainedin[I]bythemethodofbijectivecorrespondence. (See,however,
the remark there on p. 3 on the Schu¨tzenberger method.) In this paper we also
apply the Schu¨tzenberger method. However, in section 6, to obtain the generating
function of the relevant code, we return to the method of bijective correspondence
that was suggested in [I].
2. A construction of subshifts
Let there be given an irreducible finite directed labeled graph G with a dis-
◦
tinguished vertex v and with labeling alphabet Σ and labeling map λ . We
◦ ◦ ◦
assume that the labeling map λ is 1-right resolving. It is allowed that G is the
◦ ◦
degenerate graph with the one vertex v and no edges. Σ is allowed to contain
◦ ◦
only one symbol. G can present a topological Markov shift, or it can be the Fis-
◦
cher automaton of a topologically transitive sofic system. Let there also be given
SUBSHIFTS FROM SOFIC SHIFTS AND DYCK SHIFTS 5
K−,K+ ∈N,γ ∈Γ. WeconstructalabeleddirectedgraphG(λ ,v ,(K−,K+) )
γ γ ◦ ◦ γ γ γ∈Γ
from G by attaching to the vertex v directed loops that we name
◦ ◦
e(γ(−),k−), 1≤k− ≤K−,
γ γ γ
e(γ(+),k+), 1≤k+ ≤K+,
γ γ γ
and that we label by themselves,
λ (e(γ(−),k−))=e(γ(−),k−), 1≤k− ≤K−,
◦ γ γ γ γ
λ (e(γ(+),k+))=e(γ(+),k+), 1≤k+ ≤K+, γ ∈Γ.
◦ γ γ γ γ
Setting
ϕ(σ )=1, σ ∈Σ ,
◦ ◦ ◦
and
ϕ(e(γ(−),k−))=γ(−), 1≤k− ≤K−,
ϕ(e(γ(+),k+))=γ(+), 1≤k+ ≤K+,
we calla path e ,1≤i≤I,I ∈N, inthe graphG(λ ,v ,(K−,K+) ) admissible
i ◦ ◦ γ γ γ∈Γ
if (ϕ(λ(e )) is an admissible word of the Motzkin shift. We define a subshift
i 1≤i≤I
X(G(λ ,v ,(K−,K+) )) with alphabet
◦ ◦ γ γ γ∈Γ
Σ ∪{e(γ(−),k−):1≤k− ≤K−}∪{e(γ(+),k+):1≤k+ ≤K+}
◦ γ γ γ γ γ γ
by declaring a word as admissible for X(G(λ ,v ,(K−,K+) )) if it is the label
◦ ◦ γ γ γ∈Γ
sequence of an admissible path in the graph G(λ ,v ,(K−,K+) ). We say that
◦ ◦ γ γ γ∈Γ
an admissible word of X(G(λ ,v ,(K−,K+) )) is literal-non-positive (literal-
◦ ◦ γ γ γ∈Γ
non-negative)iftheimagesunderϕofallofitsymbolsareinD−∪{1}({1}∪D+).
N N
We say that a set of admissible words of X(G(λ ,v ,(K−,K+) )) is literal-
◦ ◦ γ γ γ∈Γ
uniform if all of its words are either literal-non-positive or literal-non-negative. If
the labeling λ is bijective then X(G(λ ,v ,(K−,K+) )) is a D -presentation
◦ ◦ ◦ γ γ γ∈Γ N
of X(G(λ ,v ,(K−,K+) )), that uses on G(λ ,v ,(K−,K+) ) the labeling
◦ ◦ γ γ γ∈Γ ◦ ◦ γ γ γ∈Γ
λ that labels the edges of G with 1 and sets the label equal to the value of ϕ
◦
for the other edges. Let R be the set of paths in G that start and end at the
◦
vertex v . Let C be the circular code of paths (e ) ,I ∈ N, in the graph
◦ i 1≤i≤I
G(λ ,v ,(K−,K+) ) that start and end at the vertex v and that are such
◦ ◦ γ γ γ∈Γ ◦
that the word (ϕ(λ(e )) is in the Motzkin code. Let C− be the circular code
i 1≤i≤I
that contains the paths in G(λ ,v ,(K−,K+) ) that are concatenations of an
◦ ◦ γ γ γ∈Γ
admissible path with a literal-non-positive label sequence and a path in C, let C0
be the circular code that contains the paths in G(λ ,v ,(K−,K+) ) that are
◦ ◦ γ γ γ∈Γ
concatenations of a path in C and of a path in R, and let C+ be the circular code
that contains the paths in G(λ ,v ,(K−,K+) ) that are concatenations of a
◦ ◦ γ γ γ∈Γ
path in C and of an admissible path with a literal-non-negative label sequence.
We set
K = K−, K = K+,
− γ + γ
γ∈Γ γ∈Γ
X X
K = K−K+.
γ γ
γ∈Γ
X
Lemma 2.1.
g(C0,z)= 1(1− 1−4Kg(R,z)2z2).
2
p
6 KOKOROINOUEANDWOLFGANGKRIEGER
Proof. It is
g(C0)=g(C)g(R),
and there is a counting argument (Schu¨tzenberger method) that translates a set
equation into the equation
Kg(R,z)z2
g(C,z)= . (cid:3)
1−g(R,z)g(C,z)
Denote by D− the circular code of paths in G(λ ,v ,(K−,K+) ) that are
◦ ◦ γ γ γ∈Γ
obtained by letting one of the edges e(γ(−),k−),1≤k− ≤K−, follow a path in R
γ γ γ
anddenotebyD+ bethecircularcodeofpathsinG(λ ,v ,(K−,K+) )thatare
◦ ◦ γ γ γ∈Γ
obtained by letting a path in R follow one of the edges e(γ(+),k+),1≤k+ ≤K+.
γ γ γ
Lemma 2.2.
g(C0,z) g(C0,z)
g(C−,z)= , g(C+,z)= .
1−K g(R,z)z 1−K g(R,z)z
− +
Proof. EverypathinC− canbe writtenuniquely asa concatenationofapaththat
isaconcatenationofpathsinD− andofapaththatisaconcatenationofapathin
R and a path in C, and, symmetrically, every path in C+ can be written uniquely
as a concatenationof a path that is a concatenationof paths in C and a path in R
and of a path that is a concatenation of paths in D+. Apply the Schu¨tzenberger
method. (cid:3)
Denote by D− the circular code of words with a last symbol e(γ(−),k−),1 ≤
◦ γ
k− ≤ K−, that follows the label sequence of a path in R and denote by D+ be
γ γ ◦
the circular code of words with a first symbol e(γ(+),k+),1 ≤ k+ ≤ K+ that
γ γ γ
is followed by the label sequence of a path in R. Denote the sofic shift that is
presented by G by Y0 and denote by Y−(Y+) the coded system of D−(D+). Y−
◦ ◦ ◦
andY+ aresoficsystems. DenotetheadjacencymatrixofG byA . Alsodenote
◦ G◦
by(1−AG◦z)hv◦i the matrix thatis obtainedby deleting inthe matrix(1−AG◦z)
the v -th row and the v -th column.
◦ ◦
Theorem 2.3.
g(R,z)= det(1−AG◦z)hv◦i, (2.1)
det(1−A z)
G◦
ζ (z)=
X(G(λ◦,v◦,(Kγ−,Kγ+)γ∈Γ)
2ζ (1+ 1−4Kg(R,z)2z2)
Y0
.
(1−2K g(R,z)z+ 1−4Kg(R,z)2z2)(1−2K g(R,z)z+ 1−4Kg(R,z)2z2)
− p +
Proof. (2.1)follows frpom anapplicationof Cramer’srule. Y−p,Y+, andC−,C0,C+,
satisfy the relations (1.3-6). One has here
Y−∩Y+ =Y0.
and also
P(Y−)=P(D−)∪P(Y0), P(Y+)=P(Y0)∪P(D+).
◦ ◦
Oneobtains the theorembymeans ofLemma 2.1andLemma2.2 from(1.7)where
oneuses,thatthe 1-rightresolvingpropertyofthe labeling λ implies thatthere is
◦
aone-to-onecorrespondencebetweenthe pathsinR andtheir labelsequences. (cid:3)
SUBSHIFTS FROM SOFIC SHIFTS AND DYCK SHIFTS 7
Corollary 2.4. Under the assumption that K ≥ K , the topological entropy of
− +
X(G(λ ,v ,(K−,K+) )isequalofthenegativelogarithm ofthesmallestpositive
◦ ◦ γ γ γ∈Γ
root of
K
−
zg(R,z)= .
K2 +K
−
Proof. One checks that
1
limsup logcard P (X(G(λ ,v ,(K−,K+) ))=
n n ◦ ◦ γ γ γ∈Γ
n→∞
1
lim logcard L (X(G(λ ,v ,(K−,K+) )),
n→∞n n ◦ ◦ γ γ γ∈Γ
and one applies Theorem 2.1 and Theorem 2.3. (cid:3)
We see the Dyck shift D here as the special case Y = ∅,g(R) = 1, K− =
N γ
K+ = 1,γ ∈ Γ. Also, as an example for G take a bouquet G (J,Q) of J circles,
γ ◦ (cid:13)
each circle made up of Q edges, J,Q ∈N: Besides the vertex v ,G has vertices
(cid:13) (cid:13)
v ,1 ≤ j ≤ J,1 ≤ q < Q, and there is an edge from the vertex v to each of the
j,q (cid:13)
verticesv ,1≤j ≤J,thereisanedgefromthevertexv tothevertexv ,1≤
j,1 j,q j,q+1
j ≤ J,1 ≤ q < Q−1, and an edge from each of the vertices v ,1 ≤ j ≤ J, to
j,Q−1
v . Withλ (J,Q)thelabelingmapthatlabelstheedgesofG bythemselveswe
(cid:13) (cid:13) (cid:13)
writeG (N,J,Q)forG(λ (J,Q),v ,(1,1) ). G (N,1,1)is the Motzkinshift
(cid:13) (cid:13) (cid:13) γ∈Γ (cid:13)
M . ThesubshiftsX(G (N,J,1))wereintroducedin[I]wheretheirzetafunction
N (cid:13)
[I, Proposition 2.3] and topological entropy [I, Corollary 2.2] were determined.
Corollary 2.5.
2(1−JzQ+ (1−JzQ)2−4Nz2)
ζ (z)= , N >1.
X(G(cid:13)(N,J,Q)) (1−JzQ−2Nz+ (1−JzQ)2−4Nz2)2
p
Proof. In this case p
1
g(R,z)= . (cid:3)
1−JzQ
Corollary 2.6. The topological entropy of X(G (N,J,Q))is equal to the negative
(cid:13)
logarithm of the positive root of
N +1 1
zQ+ z− =0.
J J
Proof. See Corollary 2.4. (cid:3)
We note another special case.
Corollary 2.7.
2(1−z2+ (1−z2)2−4Nz2)
ζ (z)= , N >1.
X(G(cid:13)(N,1,2)) (1−z2−2Nz+ (1−z2)2−4Nz2)2
p
Corollary 2.8. p
h(X(G (N,1,2)))=log2−log( (N +1)2+4−N −1), N >1.
(cid:13)
p
8 KOKOROINOUEANDWOLFGANGKRIEGER
As another example take for G the Fischer automaton of the even system [Ki,
◦
Section 6.1] There are two vertices v and v , the labeling alphabet is {0,1},
even odd
and there are edges from v to v and from v to v that are assigned by
even odd odd even
alabelingmapλ thelabel0,andthereisaloopatv thatisassignedbythe
even even
labeling map λ the label 1. X(G(λ ,v ,(1,1) )) is the Schr¨oder shift
even even even γ∈Γ
Sch ,N >1.
N
Corollary 2.9.
2(1+z)(1−z−z2+ (1−z−z2)2−4Nz2)
ζ (z)= , N >1.
SchN (1−(2N +1)z−z2+ (1−z−z2)2−4Nz2)2
p
Proof. In this case by (2.1) p
1
g(R,z)= . (cid:3)
1−z−z2
Corollary 2.10.
h(Sch )=log2−log( (N +2)2+4−N −2), N >1.
N
Proof. See Corollary 2.4. p (cid:3)
Corollary 2.11.
ζ (z)=
X(G(λeven,vodd,(1,1)γ∈Γ))
2(1+z)(1−z−z2+ (1−z−z2)2−4Nz2(1−z)2)
, N >1.
(1−(2N +1)z+(2N −1)z2+ (1−z−z2)2−4Nz2(1−z)2)2
p
Proof. In this case by (2.1) p
1−z
g(R,z)= . (cid:3)
1−z−z2
Corollary 2.12.
h(X(G(λ ,v ,(1,1) )))=log2+logN−log(N+2− (N +2)2−4N),
even odd γ∈Γ
p N >1,
Proof. See Corollary 2.4. (cid:3)
3. Assigning to the elements of a finite set subsets of the set
Let Ψ be a map that assigns to a γ ∈Γ a non-empty subset Ψ(γ) of Γ. We say
that a permutation π of Γ is a symmetry of Ψ if
π(Ψ(γ))=Ψ(π(γ)), γ ∈Γ.
We introduce an equivalence relation ∼ into the set Γ where for α,β ∈ Γ,α ∼ β
means that for some L ∈ N there are γ ∈ Γ,0 ≤ l ≤ L,γ = α,γ = β, such that
l 0 L
Ψ(γ )=Ψ(γ )orthereexistsasymmetryπofΨsuchthatπ(γ )=γ ,0<l≤L.
l−1 l l−1 l
The set
∆ =Ψ−1(Γ)
Γ
SUBSHIFTS FROM SOFIC SHIFTS AND DYCK SHIFTS 9
is an ∼-equivalence class. We denote by ∆ the set of elements of Γ that are
\
∼-equivalent to a γ ∈Γ such that
Ψ(γ)=Γ\{γ},
andwedenoteby∆ the setofelementsofΓthatare∼-equivalenttoaγ ∈Γsuch
•
that
Ψ(γ)={γ}.
We set
∆ =Γ\(∆ ∪∆ ∪∆ ),
◦ Γ \ •
and we set
N =card(∆ ),N =card(∆ ),N =card(∆ ),
Γ Γ \ \ • •
N =N −(N +N +N ).
◦ Γ \ •
We denote the set of ∼-equivalence classes in ∆ by A.
◦
Lemma 3.1.
card(Ψ(α)∩∆ )=card(Ψ(α′)∩∆ ),
Γ Γ
card(Ψ(α)∩∆ )=card(Ψ(α′)∩∆ ),
\ \
card(Ψ(α)∩∆ )=card(Ψ(α′)∩∆ ), α,α′ ∈A∈A.
• •
Proof. For a symmetry π of Ψ,
card(Ψ(γ)∩∆ )=card(Ψ(π(γ))∩∆ ),
Γ Γ
card(Ψ(γ)∩∆ )=card(Ψ(π(γ))∩∆ ),
\ \
card(Ψ(γ)∩∆ )=card(Ψ(π(γ))∩∆ ), γ ∈Γ. (cid:3)
• •
In view of Lemma 3.1 we can introduce the notation
K (Γ)=card(Ψ(α)∩∆ ),
A Γ
K (\)=card(Ψ(α)∩∆ ),
A \
K (•)=card(Ψ(α)∩∆ ), α∈A∈A.
A •
Lemma 3.2.
card(Ψ(α)∩B)=card(Ψ(α′)∩B), α,α′ ∈A∈A,B ∈A.
Proof. For a symmetry π of Ψ,
card(Ψ(γ)∩B)=card(Ψ(π(γ)∩B), γ ∈Γ,B ∈A. (cid:3)
In view of Lemma 3.2 we can introduce the notation
K (B)=card(Ψ(α)∩B), α∈A∈A,B ∈A.
A
10 KOKOROINOUEANDWOLFGANGKRIEGER
4. Excluding a literal-uniform set of words of length two from
Dyck shifts
We continue to consider a mapping Ψ that assigns to γ ∈Γ a non-empty subset
Ψ(γ)ofΓ. Denote by X the subshift that is obtainedby removingfromthe Dyck
Ψ
shift D the words
N
β(+)α(+), α∈Γ,β ∈Γ,β ∈/ Ψ(α).
Denote by D the language of words in the Dyck code that begin with γ(−) and
γ
that are admissible for X . We set
Ψ
ξ =1− g(D ).
γ
γ∈Γ
X
Lemma 4.1.
1
g(D ,z)=z2(1+ g(D ,z)), α∈Γ.
α β
ξ(z)
β∈XΨ(α)
Proof. Apply the Schu¨tzenberger method. (cid:3)
Lemma 4.2. Let 1≤K ≤N, and let
card(Ψ(γ))=K, γ ∈Γ.
Then there exists for α,β ∈Γ a length preserving bijection
η(α,β):D →D .
α β
Proof. Denote
h(γ(−))=1,h(γ(+))=−1 γ ∈Γ,
and for a Dyck code word (d ) set
q 1≤q≤Q
H(d)= max h(d ).
q
1≤q<Q
1≤r≤q
X
For H ∈ N denote by D (α) the set of d ∈ D such that H(d) = H. Choose for
H α
α,β ∈Γ bijections
χ(α,β):Ψ(α)→Ψ(β).
One constructs inductively length preserving bijections
η (α,β):D (α)→D (β), H ∈N,
H H H
where one sets
η (α,β)(α(−)α(+)) =β(−)β(+), α,β ∈Γ.
1
Assume that for α,β ∈Γ and for H ∈N, the bijections
η (α,β): D (α)→D (β), 1≤H ≤H,
H H H
have been constructed. Then let η (α,β) be the map that carries a word
H+1
e
e α(−)(de(l)) α(e+)∈D (α),
1≤l≤L H+1
where
d(l) ∈D0,H(d(l))≤H, 1≤l ≤L∈N,
with γ ∈Ψ(α) given by d(L) ∈D , into the word
γ
β(−)(d(l)) η (γ,χ(α,β)(γ))(d(L))β(+).
1≤l<L H(d(L))